Convex isoperimetric sets in the Heisenberg group

(1)

Convex isoperimetric sets in the Heisenberg group

ROBERTOMONTI ANDMATTHIEURICKLY

Abstract. We characterize convex isoperimetric sets in the Heisenberg group.

We first prove Sobolev regularity for a certain class ofR²-valued vector fields of bounded variation in the plane related to the curvature equations. Then we show that the boundary of convex isoperimetric sets is foliated by geodesics of the Carnot-Carath´eodory distance.

Mathematics Subject Classification (2000): 49Q20 (primary); 53C17 (secondary).

1.

Introduction

We identify the Heisenberg group

H

¹^with

C

×

R

endowed with the group law (z, t)(z, t) =

z+ z, t + t+ 2 Im (z¯z) ,

where t, t∈

R

^{and z}= x + iy, z= x+ iy ∈

C

. The Lie algebra of left-invariant vector fields is spanned by

X = ∂

∂x + 2y∂

∂t, Y = ∂

∂y − 2x ∂

∂t and T = ∂

∂t,

and the distribution of planes spanned by X and Y , called horizontal distribution, generates the Lie algebra by brackets.

The natural volume in

H

¹ is the Haar measure, which, up to a positive factor, coincides with Lebesgue measure in

R

³ =

C

×

R

. Lebesgue measure is also the Riemannian volume of the left-invariant metric for which X , Y and T are orthonor- mal. We denote by|E| the volume of a (Lebesgue) measurable set E ⊂

H

¹^{. The} horizontal perimeter (or simply perimeter) of E is

P(E)=sup

E

Xϕ1+ Y ϕ2

d xd ydt  ϕ1, ϕ2 ∈ C_c¹(

R

³), ϕ²₁+ ϕ₂²≤ 1

. (1.1) If P(E) < +∞, the set E is said to be of finite perimeter. Perimeter is left-invariant and 3-homogeneous with respect to the group of dilationsδ_λ:

H

¹→

H

¹^,δ_λ(z, t) = Received March 31, 2008; accepted October 14, 2008.

(2)

(λz, λ²t), λ > 0, that is P(δ_λ(E)) = λ³P(E). Definition (1.1) of perimeter is mod- elled on De Giorgi’s notion of perimeter in Euclidean spaces which was generalized in [10] to Carnot-Carath´eodory spaces. For smooth sets (e.g. of class C²), perimeter coincides with the Minkowski content and with the 3-dimensional Hausdorff measure of∂ E constructed by means of the standard Carnot-Carath´eodory metric in

H

¹ (see [16] and [9], respectively). Volume and perimeter are related via the isoperimetric inequality

|E| ≤ CisopP(E)⁴^/3, (1.2)

where C_isop > 0 and E ⊂

H

¹ is any measurable set with finite perimeter and volume. This inequality is proved by P. Pansu in [17] and [18] for smooth domains with the 3-dimensional Hausdorff measure of ∂ E in

H

¹ replacing P(E). In the general form (1.2), the inequality is proved in [10] (see also [8]). The problem of computing the sharp constant C_isop leads to the notion of isoperimetric set. A set E ⊂

H

¹ ^{with 0} < |E| < +∞ is an isoperimetric set if it minimizes the isoperimetric ratio

Isop(E) = P(E)⁴^/3

|E| . (1.3)

Isoperimetric sets do exist: this is proved in [12] by a concentration-compactness argument. Pansu notes that the boundary of a smooth isoperimetric set has “constant mean curvature” and that a smooth surface has “constant mean curvature” if and only if it is foliated by horizontal lifts of plane circles with constant radius. Then he conjectures that an isoperimetric set is obtained by rotating around the center of the group a geodesic joining two points in the center. Recently, Pansu’s conjecture reappeared in [11].

The problem of determining isoperimetric sets is interesting for two reasons.

On the one hand, the non commutative group law makes it difficult to prove by a rearrangement argument that the isoperimetric ratio (1.3) is minimized by rotationally symmetric sets, as it is natural to conjecture. On the other hand, there is no regularity theory for measurable sets in

H

¹minimizing perimeter with (or without) a volume constraint. So, new techniques and ideas are needed.

All known results assume either symmetry or regularity (or both). In fact, assuming rotational symmetry and regularity it is easy to determine the isoperimetric profile (see [13], and [22] for the general case). Actually, it suffices to assume the rotational symmetry of a certain horizontal section (see [7] and espe- cially the calibration argument in [21]). The solution of the rotationally symmetric case with no regularity assumption is in [14]. On the other hand, isoperimetric sets can be also determined assuming only the C² regularity of the boundary and no symmetry (see [23]). Further evidence supporting Pansu’s conjecture is provided by [15], where a 2-dimensional version of the problem is solved. We refer to the monograph [3] for a more detailed introduction to the isoperimetric problem in the Heisenberg group.

In this article, we characterize isoperimetric sets which are convex. By convex set we mean a subset of

H

¹ =

R

³ which is convex with respect to the standard

(3)

vector space structure of

R

³. Convexity is a left invariant property in

H

¹ ^because left translations are affine mappings.

Theorem 1.1 (Convex isoperimetric sets). Up to a left translation and a dilation, any closed convex isoperimetric set in

H

¹coincides with

E_isop=

(z, t) ∈

H

¹ |t| ≤ arccos |z| + |z|

1− |z|², |z| ≤ 1

. (1.4)

The boundary of the set in (1.4) is foliated by Heisenberg geodesics, it is globally of class C², but it fails to be of class C³at the north and south poles(0, ±π/2). At these points, the plane spanned by the vector fields X and Y , the horizontal plane, is tangent to the boundary.

We explain the main steps in the proof of Theorem 1.1. Let us consider a bounded convex set of the form

E =

(z, t) ∈

H

¹| z ∈ D, f (z) ≤ t ≤ g(z)

, (1.5)

where D ⊂

R

²is a compact convex set in the plane with nonempty interior, and

−g, f : D →

R

are convex functions. If E is isoperimetric, then the function f : D →

R

satisfies the partial differential equation

div

∇ f (z) + 2z^⊥

|∇ f (z) + 2z^⊥|

= 3P(E)

4|E| (1.6)

in int(D) \ ( f ). Here and in the following we let z = (x, y) and z^⊥ = (−y, x).

We call the singular set( f ) =

z ∈ int(D) | − 2z^⊥ ∈ ∂ f (z)

the characteristic set of f . This set is always contained in a line segment. The basic facts concerning

( f ) are discussed in the appendix. Equation (1.6) is derived in Section 2.

The curvature operator in (1.6) has been studied by several authors under dif- ferent regularity assumptions (besides the previous references, see also [4–6, 19, 20]). In the smooth case, the number

H = 3P(E)

4|E| (1.7)

is known as the (horizontal) curvature of∂ E. In our case, equation (1.6) is to be interpreted in distributional sense. The distributional derivatives of u(z) = ∇ f (z)+

2z^⊥ are measures, because f is a convex function, and so the equation states that the distributional divergence of u/|u| is constant.

Our goal is to give equation (1.6) a pointwise meaning along integral curves of the vector field orthogonal to u/|u|. The first step is to show that the equation holds in the Sobolev sense. Precisely, we prove that the distributional derivative of u/|u|

is a measure which is absolutely continuous with respect to Lebesgue measure.

This result is a corollary of the following regularity theorem for BV vector fields, which is interesting in itself. We prove a slightly more general statement in the third section.

(4)

Theorem 1.2 (Improved regularity). Let ⊂

R

²be a bounded open set and let u∈ BV (;

R

²) be a vector field. Suppose that:

(i) There existsδ > 0 such that |u(z)| ≥ δ for

L

²^{-a.e. z}∈ ;

(ii) div u^⊥ ∈ L¹();

(iii) div u

|u|

∈ L¹().

Then we have u/|u| ∈ W¹^,1(;

R

²).

The improved regularity of the boundary is the starting point for the geometric characterization of convex isoperimetric sets. The vector fieldv(z) = −u^⊥(z) = 2z−∇ f^⊥(z) belongs to BVloc(int(D);

R

²). Moreover, its distributional divergence is in L^∞, in fact

divv = 4 in int(D). (1.8)

Thanks to the theory on the Cauchy Problem for BV vector fields recently devel- oped by Ambrosio in [1], the bound on the divergence ensures the existence of a unique regular Lagrangian flow  : K × [− , ] → D starting from a compact set K ⊂ int(D), where > 0 is small enough. For

L

²^{-a.e. z} ∈ K , the curve s → (z, s) is an integral curve of v passing through z at time s = 0. In Section 4, we show that

L

²-a.e. integral curve ofv is an arc of circle.

Theorem 1.3 (Foliation by circles). Let E ⊂

H

¹be a convex isoperimetric set of the form (1.5) and let K ⊂ int(D)\( f ) be a compact set. Then, for

L

²^{-a.e. z} ∈ K , the curve s → (z, s) is an arc of circle having radius 1/H, with H > 0 as in (1.7).

The proof of Theorem 1.3 relies upon a reparameterization argument. The vector field v has a regular flow starting from K ⊂ int(D) \ ( f ) but it is only in BV_loc(int(D);

R

²). On the other hand, v/|v| is in W_loc¹^,1(int(D);

R

²), but its divergence is only in L¹_loc(int(D)) and so we have no regular flow for v/|v|. In order to compute the second order derivative of a generic integral curve of v, we introduce a suitable reparameterizationγ (s) = (z, τ(s)). We show that for any vector field w ∈ W¹^,1(;

R

²) defined in some open neighborhood of K , the curveκ(s) = w(γ (s)) is in W¹^,1for

L

²^{-a.e. z} ∈ K and moreover

˙κ = (∇w ◦ γ ) ˙γ in the weak sense. (1.9) Using the chain rule (1.9) with w = v/|v|, which is in W¹^,1 by Theorem 1.2, equation (1.6) can be given a pointwise meaning along the flow, and the integral curves ofv turn out to be arcs of circles.

Since the horizontal lift of the flow  foliates the graph of the function f , it follows that the bottom (and upper) part of the boundary of E is foliated by geodesics of the Heisenberg Carnot-Carath´eodory metric. However, this is not yet

(5)

enough to finish our argument. We still need to show that the bottom and upper parts of∂ E match together in the proper way. To do this, we write E in the form

E=

(x, y, t) ∈

H

¹| (y, t) ∈ F, h(y, t) ≤ x ≤ k(y, t)

, (1.10) for some compact convex set F ⊂

R

² and convex functions h, −k : F →

R

^{. The} analysis carried out for f can be also carried out for h even though in this case computations are, unfortunately, more complicated. This provides the last piece of information needed to get the set E_isopin (1.4).

A short overview is now in order. In Section 2, we derive the curvature equations for convex isoperimetric sets. In Section 3, we prove the generalized version of Theorem 1.2 which is needed in Section 4, where we establish the foliation by geodesics property for convex isoperimetric sets. Finally, in Section 5 we prove Theorem 1.1. The results concerning convex sets are collected in the appendix at the end of the paper.

2.

Curvature equations for convex isoperimetric sets

We derive partial differential equations for certain vector fields built from the functions which parameterize the boundary of convex isoperimetric sets. We study graphs of the form t = f (x, y) and graphs of the form x = h(y, t). We use the following representation formula for perimeter. If E ⊂

H

¹ is a bounded set such that∂ E is locally a Lipschitz surface, then

P(E) =

∂ E

(X · ν)²+ (Y · ν)²d

H

², (2.1)

whereν is a unit normal to ∂ E. Here and in the following, · denotes the standard inner product in

R

³ ^or

R

² (we think of X and Y as vectors in

R

³^).

H

²^{is the 2-} dimensional Hausdorff measure in

R

³with respect to the usual Euclidean distance.

For a proof of formula (2.1), see [9].

Graphs of the form t = f (x, y)

We denote elements of

H

¹ =

R

²×

R

^by(z, t) with t ∈

R

^{and z} = (x, y) ∈

R

²^. We write z^⊥ = (−y, x). Let E be a convex set in

H

¹of the form (1.5). We define the characteristic set of the convex function f : D →

R

^as

( f ) =

z ∈ int(D) | − 2z^⊥ ∈ ∂ f (z)

, (2.2)

where∂ f (z) stands for the subdifferential of f at z.

Proposition 2.1 (Curvature equation I). Let E ⊂

H

¹be a convex isoperimetric set with perimeter P(E) and volume |E|. Then the function f : D →

R

satisfies in distributional sense the partial differential equation

div

∇ f (z) + 2z^⊥

|∇ f (z) + 2z^⊥|

= H (2.3)

in int(D) \ ( f ), with H = 3P(E)/4|E|.

(6)

Proof. By Theorem A.1 in the appendix,( f ) is contained in the union of at most two line segments.

Letϕ ∈ C_c^∞(int(D) \ ( f )) and for ε ∈

R

, consider the set E_ε =

(z, t) ∈

H

¹| z ∈ D, f (z) + εϕ(z) ≤ t ≤ g(z) .

We let P(ε) = P(E_ε), V (ε) = |E_ε| and R(ε) = P(ε)⁴/V (ε)³. If E is an isoperi- metric set, then R(ε) has a minimum at ε = 0, and then

R(0) = P³ V⁴

4PV − 3PV

ε=0

= 0. (2.4)

There existsε0 > 0 such that V(ε) = −

D

ϕ(z) dz for |ε| < ε0. (2.5) Letν_εbe the exterior unit normal to∂ E_εand let S_ε =

(z, f (z)+εϕ(z)) ∈

H

¹| z ∈ D

be the graph of f + εϕ. By the Heisenberg area formula (2.1) and from the standard area formula for graphs of functions in Euclidean spaces, we find

P(ε) = d dε

S_ε

(X · ν_ε)²+ (Y · ν_ε)²d

H

²

= d dε

D

|∇ f (z) + ε∇ϕ(z) + 2z^⊥| dz.

(2.6)

By Proposition A.2 in the appendix, for any compact set K ⊂ int(D) \ ( f ), there isδ > 0 such that |∇ f (z) + 2z^⊥| ≥ δ for

L

²^{-a.e. z} ∈ K . Then we can interchange derivative and integral for all smallε. At ε = 0 we obtain

P(0) =

D

∇ f (z) + 2z^⊥

|∇ f (z) + 2z^⊥| · ∇ϕ(z) dz. (2.7) From (2.4), (2.5) and (2.7) we find

4|E|

D

∇ f (z) + 2z^⊥

|∇ f (z) + 2z^⊥|· ∇ϕ(z) dz = −3P(E)

Dϕ(z) dz for arbitraryϕ ∈ C_c^∞(int(D) \ ( f )). This is (2.3).

Graphs of the form x = h(y, t)

We denote points in

H

¹ =

R

×

R

² ^by(x, ζ) with x ∈

R

^andζ = (y, t) ∈

R

²^. Let E be a convex set in

H

¹of the form (1.10). We define the characteristic set of the convex function h : F →

R

^{as the set}(h) of the points ζ ∈ F such that the horizontal plane spanned by the vector fields X and Y at the point p= (h(ζ ), ζ) ∈

R

×

R

²is a supporting plane for E (i.e. the plane does not intersect the interior of E). By Theorem A.1 in the appendix,(h) is contained in the union of at most two line segments.

(7)

Proposition 2.2 (Curvature equation II). Let E ⊂

H

¹be a convex isoperimetric set with perimeter P(E) and volume |E|. Then the function h : F →

R

satisfies in distributional sense the partial differential equation

(∂y− 2h∂t) u₁

|u|

− 2y∂t

u₂

|u|

= H (2.8)

in int(F)\(h), with H = 3P(E)/4|E| and u = (u1, u2) = (hy−2hht, 1−2yht).

Proof. Notice that u ∈ BVloc(int(F);

R

²) and moreover, by Proposition A.3 in the appendix, for each compact set K ⊂ (int(F) \ (h)), there exists δ > 0 such that

|u| ≥ δ

L

²-a.e. in K . Hence u/|u| ∈ BVloc(int(F) \ (h);

R

²).

Letϕ ∈ C_c^∞(int(F) \ (h)) and for any ε ∈

R

consider the set E_ε =

(x, ζ) ∈

H

¹| ζ ∈ F, h(ζ) + εϕ(ζ) ≤ x ≤ k(ζ ) .

We let P(ε) = P(E_ε), V (ε) = |E_ε| and R(ε) = P(ε)⁴/V (ε)³. Denoting by S_ε =

(h(ζ) + εϕ(ζ ), ζ) ∈

H

¹ | ζ ∈ F

the graph of h+ εϕ and by ν_εthe exterior unit normal to∂ E_ε, from the Heisenberg area formula (2.1) and from the standard area formula we get

P(ε)= d dε

S_ε

(X · ν_ε)²+ (Y · ν_ε)²d

H

²

= d dε

F

1− 2y(ht+εϕt)2

+

(hy+εϕy) − 2(h+εϕ)(ht+εϕt)2₁_/2 dζ.

Because we have(hy− 2hht)²+ (1 − 2yht)²≥ δ²> 0

L

²-a.e. in a neighbourhood of spt(ϕ), we can differentiate under the integral sign and we obtain

P(0) =

F

(hy− 2hht)(ϕy− 2(ϕh)t) − 2yϕt(1 − 2yht)

(hy− 2hht)²+ (1 − 2yht)² dζ

atε = 0. An integration by parts yields P(0) = −

F

ϕ d

(∂y− 2h∂t) u₁

|u|

− 2y∂t

u₂

|u|

,

where the derivatives of u/|u| are measures. If E minimizes the isoperimetric ratio then, as in the proof of Proposition 2.1, we get

4|E|

F

ϕ d

(∂y− 2h∂t) u₁

|u|

− 2y∂t

u₂

|u|

= 3P(E)

F

ϕ dζ

for arbitraryϕ ∈ C_c^∞(int(F) \ (h)). This is (2.8).

(8)

3.

Improved regularity of the boundary

In this section, we prove a regularity result for vector fields with bounded variation in the plane arising from the parameterization of the boundary of convex isoperimetric sets.

Let ⊂

R

²be an open set and let a, b ∈ C() be continuous functions. We consider the differential operator

M

acting on BV(;

R

²) defined by

M

^u= (∂1− a∂2)u1+ b∂2u₂. (3.1) In general,

M

u is a measure in. We have

M

= div when a = 0 and b = 1.

When a = 2h and b = −2y,

M

is the operator appearing in the left hand side of (2.8).

We recall that a vector field u ∈ L¹(;

R

²) has approximate limit ¯u(q) ∈

R

² at the point q ∈ if

limr↓0−

B(q,r)|u − ¯u(q)| d

L

²= 0. (3.2)

Here, B(q, r) denotes the open Euclidean ball of radius r centered at q. The ap- proximate discontinuity set of u is the set S_u of points in at which u has no approximate limit. The jump set of u is the set J_u of points q ∈ for which there exist u⁺(q), u⁻(q) ∈

R

²^andνu(q) ∈ S¹such that u⁺(q) = u⁻(q) and

lim

r↓0−

B^±(q,r)|u − u^±(q)| d

L

²= 0, (3.3)

where B^±(q, r) =

q ∈ B(q, r) | ± (q− q) · νu(q) > 0

. Finally, the precise representative u^∗ : →

R

²^{of u is:}

u^∗(q) =











¯u(q) q ∈ \ Su,

1 2

u⁺(q) + u⁻(q)

q ∈ Ju,

0 q ∈ Su\ Ju.

(3.4)

By the Lebesgue density theorem, it is u= u^∗

L

²^{-a.e. in}.

Theorem 3.1 (Improved regularity). Let ⊂

R

²be a bounded open set, let u= (u1, u2) ∈ BV (;

R

²), and let a, b ∈ C() be continuous functions such that b = 0 in . Assume that:

(i) There existsδ > 0 such that |u| ≥ δ

L

²^{-a.e. in};

(ii)

M

^u^⊥∈ L¹(), where u^⊥= (−u2, u1);

(iii)

M

u

|u|

∈ L¹().

(9)

Then u/|u| ∈ W¹^,1(;

R

²) and there exists a function µ : Ju → (0, +∞), such that u⁻ = µu⁺on J_u.

Proof. If u ∈ BV (;

R

²) the measure Du has the decomposition Du = D^au+ D^su with D^au

L

²^{and D}^s^u ⊥

L

²^{. D}^a^u= ∇u

L

²is the absolutely continuous part of the measure, with∇u ∈ L¹(; M²^×2) and M²^×2denotes the space of 2× 2 matrices with real entries. Moreover, it is D^su = D^cu + D^ju, where D^cu = D^su  \ Su is the Cantor part and D^ju = D^su S_u is the jump part. By the representation theorem of Federer–Vol’pert and by Alberti’s rank one theorem, the measures D^cu and D^ju admit the representations

D^ju= (u⁺− u⁻) ⊗ νu

H

¹ ^Ju and D^cu= η ⊗ ξ |D^cu|, (3.5) where |D^cu| is the total variation of D^cu and η, ξ : → S¹are suitable Borel maps. The measure|D^cu| is absolutely continuous with respect to

H

¹. The set J_u is an

H

¹-rectifiable Borel subset of S_uwith

H

¹(Su\ Ju) = 0. For these and related results on BV vector fields we refer the reader to the monograph [2].

We claim that both the jump and the Cantor parts of D(u/|u|) vanish. Let F:

R

²→

R

²be a smooth mapping such that

F(0) = 0, F(q) = q

|q| for |q| ≥ δ

2 and |∇ F| ∈ L^∞(

R

²). (3.6) If|q| > δ/2, the derivative of F is

∇ F(q) = 1

|q|³q^⊥⊗ q^⊥. (3.7)

By (i), the vector fieldv = F ◦ u is in BV (;

R

²) (this fact is implicitly assumed in (iii)).

By the chain rule for BV vector fields, we have

D^av = ∇ F(u^∗)∇u

L

² ^and ^D^cv = (∇ F(u^∗)η) ⊗ ξ |D^cu|. (3.8) A point q ∈ belongs to J_v if and only if q ∈ Ju and F(u⁺(q)) = F(u⁻(q)).

Moreover, it is D^jv =

F(u⁺) − F(u⁻)

⊗ νu

H

¹ ^Ju. Notice that|u^±(q)| ≥ δ for all q ∈ Ju, by (i). Then, the jump set ofv is

J_v =

q∈ Ju | u⁺(q)/|u⁺(q)| = u⁻(q)/|u⁻(q)|

(3.9) and the jump part of Dv is

D^jv = u⁺

|u⁺| − u⁻

|u⁻|

⊗ νu

H

¹ ^Ju. (3.10)

(10)

By assumption (ii), the part of the measure−∂1u₂+ b∂2u₁+ a∂2u₂concentrated on J_u vanishes. From the formula for D^ju in (3.5), we can compute D_k^ju_l for k, l = 1, 2, and we obtain

(u⁻₂ − u⁺₂), b(u⁺₁ − u⁻₁) + a(u⁺₂ − u⁻₂)

· νu= 0 (3.11)

H

¹^{-a.e. on J}u.

By assumption (iii), the part of the measure(∂1− a∂2)(u1/|u|) + b∂2(u2/|u|) concentrated on J_v vanishes. Thus, using (3.10), we can compute D_k^j(ul/|u|) for k, l = 1, 2, and we get

u⁺₁

|u⁺|− u⁻₁

|u⁻|,au⁻₁ − bu⁻₂

|u⁻| −au⁺₁ − bu⁺₂

|u⁺|

· νu = 0 (3.12)

H

¹^{-a.e. on J}_v^.

From (3.12) and (3.11), we deduce that there existsλ ∈

R

such that the following system of equations is satisfied

H

¹^{-a.e. on J}_v^:









 u⁺₁

|u⁺| − u⁻₁

|u⁻| = −λ(u⁺₂ − u⁻₂) bu⁺₂ − au⁺₁

|u⁺| − bu⁻₂ − au⁻₁

|u⁻| = λ

b(u⁺₁ − u⁻₁) + a(u⁺₂ − u⁻₂) . By elementary linear algebra, using b = 0, we obtain the equivalent system









 u⁺₁

|u⁺| − u⁻₁

|u⁻| = −λ(u⁺₂ − u⁻₂) u⁺₂

|u⁺| − u⁻₂

|u⁻| = λ(u⁺₁ − u⁻₁)

⇔

u⁺

|u⁺|− u⁻

|u⁻|

·

u⁺− u⁻

= 0,

that is u⁻· u⁺ = |u⁻||u⁺|,

H

¹^{-a.e. on J}_v. It follows that u⁻ = µu⁺

H

¹^{-a.e. on} J_vfor someµ : J_v→ (0, +∞), and then u⁺/|u⁺| = u⁻/|u⁻|

H

¹^{-a.e. on J}_v^{. This} proves that

H

¹(J_v) = 0 and thus D^jv = 0.

By assumption (ii), the Cantor part of the measure−∂1u₂+ b∂2u₁+ a∂2u₂ vanishes. From D^cu= η ⊗ ξ |D^cu|, we can compute D^c_ku_l for k, l = 1, 2, and we find

(−η2, bη1+ aη2) · ξ = 0 (3.13)

|D^cu|-a.e. on . By assumption (iii), the Cantor part of the measure (∂1− a∂2)(u1/|u|) + b∂2(u2/|u|)

vanishes, too. Thus, lettingϑ = (ϑ1, ϑ2) = ((u^∗)^⊥ ⊗ (u^∗)^⊥)η, we can use (3.7) and (3.8) to compute D^c_k(ul/|u|) for k, l = 1, 2, and we find

ϑ1, bϑ2− aϑ1

· ξ = 0 (3.14)

(11)

|D^cu|-a.e. on . From (3.13) and (3.14), we deduce that there exists λ ∈

R

^such

that

ϑ1= −λη2

bϑ2− aϑ1= λ

bη1+ aη2

⇔

ϑ1 = −λη2

ϑ2 = λη1. Here we used b = 0. This, in turn, is equivalent with

ϑ · η =

(u^∗)^⊥⊗ (u^∗)^⊥ η

· η = 0

|D^cu|-a.e. on . Using the identity

(u^∗)^⊥ ⊗ (u^∗)^⊥ η

· η =

(u^∗)^⊥· η2

, we deduce that(u^∗)^⊥· η = 0, and thus

(u^∗)^⊥⊗ (u^∗)^⊥

η = 0 as well. By (3.7), this proves that(∇ F(u^∗)η) ⊗ ξ = 0 |D^cu|-a.e. on , whence D^cv = 0 by (3.8).

Theorem 3.1 has the following corollaries.

Corollary 3.2. Let E ⊂

H

¹ be a convex isoperimetric set and let f : D →

R

^be the function in (1.5). Then we have

∇ f (z) + 2z^⊥

|∇ f (z) + 2z^⊥| ∈ W_loc¹^,1(int(D) \ ( f );

R

²). (3.15) Proof. The vector field u(z) = (u1(z), u2(z)) = ∇ f (z) + 2z^⊥ satisfies div u^⊥ =

−4 in int(D). By Proposition A.2 in the appendix, |u| ≥ δ > 0 on a compact set in int(D) \ ( f ). Moreover, by Proposition 2.1, we have div(u/|u|) = 3P(E)/4|E|

in int(D) \ ( f ). The claim follows from Theorem 3.1 with a = 0 and b = 1.

Corollary 3.3. Let E ⊂

H

¹be a convex isoperimetric set and let h : F →

R

^be the function in (1.10). Then we have

u

|u| ∈ W_loc¹^,1(int(F) \ ({y = 0} ∪ (h));

R

²), (3.16) with u= (u1, u2) = (hy− 2hht, 1 − 2yht).

Proof. We use Theorem 3.1 with a = 2h and b = −2y. The vector field u is in B V_loc(int(F);

R

²) and satisfies

M

^u^⊥= (∂y−2h∂t)(2yht−1)−2y∂t(hy−2hht) = 2ht(1+2yht) ∈ L^∞_loc(int(F)).

From Proposition A.3 in the appendix, it follows that u/|u| ∈ BVloc(int(F)\(h)), and moreover, by Proposition 2.2, u/|u| satisfies

M

u

|u|

= (∂y− 2h∂t) u₁

|u|

− 2y∂t

u₂

|u|

= 3P(E) 4|E|

in int(F) \ (h). Now the claim follows from Theorem 3.1.

(12)

4.

Integration of the curvature equation

In this section we solve equations (2.3) and (2.8). These equations can be integrated along a regular Lagrangian flow and the solutions are suitable arcs of circle.

We recall the definition of regular flow in the plane. Let ⊂

R

²^{be an open} set and let u ∈ BV (;

R

²) ∩ L^∞(;

R

²). For q ∈ and > 0 small enough define

C_q([− , ]; )

=

γ ∈ C([− , ]; ) γ (s) = q +

_s

0

u(γ (σ)) dσ, s ∈ [− , ]

.

Let K ⊂ be a compact set. An

L

²-measurable map : K → C([− , ]; ) is a Lagrangian flow starting from K relative to u if(q) ∈ Cq([− , ]; ) for

L

²^- a.e. q ∈ K . With abuse of notation, is identified with the map : K ×[− , ] →

, (q, s) = (q)(s). The flow is said to be regular if there exists a constant m ≥ 1 such that

1

m

L

²(A) ≤

L

²((A, s)) ≤ m

L

²(A) (4.1) for all

L

²-measurable sets A⊂ K and for all s ∈ [− , ].

Theorem 4.1 (Foliation by circles I). Let E ⊂

H

¹ be a convex isoperimetric set and let f : D →

R

be the function in (1.5). Let K ⊂ int(D) \ ( f ) be a compact set and ⊂ int(D) \ ( f ) an open neighborhood of K .

For a sufficiently small > 0, there exists a (unique) regular Lagrangian flow

 : K × [− , ] → of the vector field v(z) = 2z − ∇ f^⊥(z). Moreover, for

L

²^{-a.e. z} ∈ K , the integral curve s → (z, s) is an arc of circle with radius 4|E|/3P(E) oriented clockwise.

For a convex isoperimetric set of the form (1.10) there is an analogous statement.

Theorem 4.2 (Foliation by circles II). Let E ⊂

H

¹ be a convex isoperimetric set and let h : F →

R

be the function in (1.10). Let K ⊂ int(F) \

{y = 0} ∪ (h) be a compact set and ⊂ int(F) \

{y = 0} ∪ (h)

be an open neighborhood of K . For a sufficiently small > 0, there exists a (unique) regular Lagrangian flow  : K × [− , ] → of the vector field v(ζ ) =

1− 2yht, 2yhy − 2h , ζ = (y, t) ∈ F. Moreover, for

L

²^-a.e.ζ ∈ K , the projection onto the xy-plane of the curve

s → (h((ζ, s)), (ζ, s)) ∈

H

¹, s ∈ [− , ], (4.2) is an arc of circle with radius 4|E|/3P(E) oriented clockwise.

The existence of a (unique) regular Lagrangian flow stated in Theorems 4.1 and 4.2 follows from Ambrosio’s theory on the Cauchy problem for BV vector fields.

(13)

Theorem 4.3 (Ambrosio). Let ⊂

R

²be an open set and assume that: (i) u∈ BV (;

R

²) ∩ L^∞(;

R

²);

(ii) div u∈ L^∞().

Then, for any compact set K ⊂ and for a small enough > 0, there exists a (unique) regular Lagrangian flow : K × [− , ] → starting from K relative to u.

The existence statement is Theorem 6.2 and the uniqueness statement is in [1, The- orem 6.4]. We do not need the uniqueness of the flow in our argument. Ambrosio’s theory holds more generally for non autonomous vector fields in any space dimen- sion.

The characterization of the flow in Theorems 4.1 and 4.2 is obtained on proving that a suitable reparameterization γ of a generic integral curve of the flow has a second order derivative which satisfies the equation ¨γ = −H ˙γ^⊥ in a weak sense with H = 3P(E)/4|E|.

In order to make this reparameterization argument precise, let us consider an open set  ⊂

R

² and a vector field v in satisfying the assumptions of Theo- rem 4.3, that is

v ∈ BV (;

R

²) ∩ L^∞(;

R

²) and div v ∈ L^∞(). (4.3) The functionv is defined pointwise, i.e. we choose a representative in the equiva- lence class ofv. Our results hold independently of this choice. However, there is an exceptional set of points which may a priori depend on the representative.

Given a compact set K ⊂ and a sufficiently small > 0, there exists a unique regular Lagrangian flow : K × [− , ] → starting from K relative to v. Let λ : →

R

be a measurable function such that

0< c1≤ λ ≤ c2

L

²^{-a.e. in}. (4.4) Then, for

L

²^{-a.e. q}∈ K , the curve s → λ((q, s)) is measurable and

c₁≤ λ((q, s)) ≤ c2 for a.e. s ∈ [− , ].

This follows from (4.1) by a Fubini-type argument. In fact, if N ⊂ is an

L

²^- negligible set, then ⁻¹(N) ⊂ K × [− , ] is also negligible. Thus, for

L

²^- a.e. q ∈ K , the change of parameter σq : [− , ] → [σq(− ), σq( )] defined by

σq(s) =

_s

0 λ((q, ξ)) dξ

is bi-Lipschitz, strictly increasing and admits therefore a bi-Lipschitz, strictly increasing inverseτq : [σq(− ), σq( )] → [− , ], which satisfies

˙τq(s) = 1

˙σq(τq(s)) = 1

λ((q, τq(s))) for a.e. s ∈ [σq(− ), σq( )]. (4.5)

(14)

Consequently, for

L

²^{-a.e. q}∈ K , the curve γq : [σq(− ), σq( )] → defined by

γq(s) = (q, τq(s)) (4.6)

is absolutely continuous and satisfies

˙γq(s) = v(γq(s))

λ(γq(s)) for a.e. s∈ [σq(− ), σq( )], (4.7) i.e.γq is an integral curve of the vector fieldv/λ.

Theorem 4.4 (Chain rule for integral curves). Let ⊂

R

²be an open set, K ⊂

 be a compact subset, w ∈ W¹^,1(;

R

²) and > 0 small enough. For

L

²^{-a.e. q}∈ K , the curvew ◦ γq withγq defined in (4.6) belongs to W¹^,1((σq(− ), σq( ));

R

²) and its weak derivative is(∇w ◦ γq) ˙γq.

Proof. Let{wk}k∈Nbe a sequence of smooth vector fieldswk ∈ W¹^,1(;

R

²) con- verging tow in W¹^,1(;

R

²). The map (q, s) → w((q, s)) is measurable and integrable on K × [− , ]. By Fubini’s theorem and by the bounded volume dis- tortion property (4.1) of the flow, we have

K

− |wk((q, s)) − w((q, s))| ds dq ≤m

−

(K,s)|wk(q) − w(q)| dq ds, and

K

− ∇wk((q, s)) − ∇w((q, s))

v((q, s))ds dq

≤ v_∞

−

K

|∇wk((q, s)) − ∇w((q, s))| dq ds

≤ mv_∞

−

(K,s)|∇wk(q) − ∇w(q)| dq ds.

The curve s→ wk((q, s)) is Lipschitz, for all k ∈

N

^{and for}

L

²^{-a.e. q}∈ K , with derivative s → ∇wk((q, s))v((q, s)). Passing to a subsequence and relabelling if necessary, we conclude that

klim→∞wk((q, ·)) = w((q, ·)) in W¹^,1((− , );

R

²)

for

L

²^{-a.e. q}∈K , and that the weak derivative of w((q,·)) is ∇w((q,·))v((q,·)).

Combining this observation, (4.5) and (4.7), for anyϕ ∈ Cc^∞((σq(− ), σq( ));

R

²)

(15)

we get,

_σ_q_( )

σq(− )w(γq(s)) ˙ϕ(s) ds =

− w((q, s)) ˙ϕ(σq(s)) ˙σq(s) ds

= −

− ∇w((q, s))v((q, s)) ϕ(σq(s)) ds

= −

_σ_q_( )

σq(− )∇w(γq(s)) ˙γq(s) ϕ(s) ds.

Hence w ◦ γq ∈ W¹^,1((σq(− ), σq( ));

R

²), and its weak derivative is (∇w ◦ γq) ˙γq.

Proof of Theorem 4.1. Let f : D →

R

be the convex function in (1.5). We consider the vector fields v(z) = 2z − ∇ f^⊥(z) and u(z) = v^⊥(z) = ∇ f (z) + 2z^⊥ which are both in BV_loc(int(D);

R

²) ∩ L^∞_loc(int(D);

R

²). We have

divv = 4 + fyx− fx y= 4. (4.8) If K ⊂ int(D) \ ( f ) is a compact set and 

^int(D) \ ( f ) is a suitable open neighborhood of K , the existence of a regular Lagrangian flow : K × [− , ] →

 for some > 0 is guaranteed by Theorem 4.3.

The functionλ : →

R

^,λ = |v|, satisfies 0 < c1≤ λ ≤ c2

L

²^{-a.e. in} by Proposition A.2 in the appendix. By Corollary 3.2, the vector fieldw = v/λ belongs to W¹^,1(;

R

²), and div w^⊥= H

L

²^{-a.e. in} by (2.3), with H = 3P(E)/4|E|.

Let γz : (σz(− ), σz( )) → be the integral curve of the vector field w defined in (4.6) and satisfying (4.7). We claim that γz parameterizes an arc of circle with curvature H . Since(z, ·) is a reparameterization of γz, proving the claim concludes the proof of Theorem 4.1. The following identities hold a.e. in (σz(− ), σz( )) for

L

²^{-a.e. z}∈ K :

(i) |w ◦ γz| = 1;

(ii) (w · ∂xw) ◦ γz= (w · ∂yw) ◦ γz = 0;

(iii) (div w^⊥) ◦ γz = H.

Here, we are using the fact that if F, G : →

R

^are

L

²-measurable functions such that F = G

L

²^{-a.e. in}, then it is F ◦ = G ◦ 

L

³^{-a.e. in K} × [− , ], which is a consequence of (4.1).

By Theorem 4.4, it isγz∈ W²^,1((σz(− ), σz( ));

R

²) with ¨γz= (∇w ◦γz) ˙γz. Using (ii) and (iii), we compute

¨γz· ˙γ_z^⊥= (−div w^⊥) ◦ γz= −H.

Moreover, by (i) we also have ¨γz· ˙γz = 0 a.e. in (σz(− ), σz( )). Then ¨γz= −H ˙γz^⊥

a.e. and this implies thatγzis of class C^∞. The claim follows.

(16)

Proof of Theorem 4.2. Let h : F →

R

be the convex function in (1.10). We use the variablesζ = (y, t) ∈ F. The vector field v : F →

R

²

v(ζ ) =

1− 2yht, 2yhy− 2h

, (4.9)

is in BV_loc(int(F);

R

²) ∩ L^∞_loc(int(F);

R

²), and

divv = −4ht− 2yht y+ 2yhyt = −4ht ∈ L^∞_loc(int(F)). (4.10) The vector fieldv is the projection onto the yt-plane of the vector field

(y, t) → (hy− 2hht)∂x+ (1 − 2yht)∂y+ (2yhy− 2h)∂t, (4.11) which is both horizontal and tangent to the graph of h at

H

²-a.e. point. We denote by u = (hy − 2hht, 1 − 2yht) the projection of the vector field (4.11) onto the x y-plane. The relation betweenv and u is

v =

0 1

2y −2h

u. (4.12)

In (4.12), we think ofv and u as column vectors. The function λ : →

R

^,λ = |u|, satisfies 0< c1≤ λ ≤ c2

L

²^{-a.e. in} by Proposition A.3 in the appendix.

Let K and be as in the statement of Theorem 4.2. The existence of a regular Lagrangian flow : K × [− , ] → of v follows from Theorem 4.3.

Let F ∈ C^∞(

R

²;

R

²) be a mapping which satisfies (3.6) (with c1in place of δ). We consider the vector fields v/λ and w = F ◦ u in . Then w = u/|u| a.e. in

. We have w ∈ W¹^,1(;

R

²) by Corollary 3.3. Observing that 2yhy − 2h = 2y(hy− 2hht) + 2h(2yht− 1), we also get that v/λ ∈ W¹^,1(;

R

²). We claim that

∇wv

λ = −Hw^⊥

L

²^{-a.e. in}, (4.13)

with H = 3P(E)/4|E|. Indeed, using (3.7), we compute

∇wv

λ = (∇ F ◦ u) ∇uv λ= 1

|u|⁴

u^⊥⊗ u^⊥

∇u v

= 1

|u|³

u^⊥· (∇u v) w^⊥.

On the other hand, by (2.8) we have

M

w = H in , where

M

is the differential operator appearing in (3.1) with a= 2h and b = −2y. Let B be the 2 × 2 matrix

B=

1 0

−2h −2y

. Using (3.7) we find

M

w = tr

(∇ F ◦ u)∇uB

= 1

|u|³tr

(u^⊥⊗ u^⊥)∇uB

= 1

|u|³u^⊥· (u^⊥∇uB),

(17)

where, by a short computation based on (4.12), we have u^⊥· (u^⊥∇uB) = −u^⊥· (∇u v). This ends the proof of (4.13).

Denote byγ_ζ : [σ_ζ(− ), σ_ζ( )] → the integral curve of v/λ defined in (4.6). Then the curve s → w(γ_ζ(s)) belongs to W¹^,1((σ_ζ(− ), σ_ζ( ));

R

²) for

L

²^-a.e.ζ ∈ K , and its weak derivative is equal to (∇w ◦ γ_ζ) ˙γ_ζ, by Theorem 4.4.

Identity (4.13) holds along

L

²-a.e. curveγ_ζ. Hence, by Theorem 4.4 applied tow, we have

d

ds(w ◦ γ_ζ) = −H(w^⊥◦ γ_ζ) (4.14) in the sense of weak derivatives.

The projection of the vector fieldζ = (y, t) → (h(ζ), ζ ) onto the xy-plane belongs to W¹^,1(;

R

²). Denote by κ_ζ : [σ_ζ(− ), σ_ζ( )] →

R

² the projection of the curve (h(γ_ζ), γ_ζ) onto the xy-plane. This projection is a reparame- terization of the curve in (4.2). We claim that κ_ζ parameterizes an arc of cir- cle with curvature H oriented clockwise. By Theorem 4.4, for

L

²^-a.e. ζ ∈ K we have κ_ζ ∈ W¹^,1((σ_ζ(− ), σ_ζ( ));

R

²), and a short computation shows that the weak derivative of κ_ζ is ˙κ_ζ = w ◦ γ_ζ. From (4.14), we deduce that κ_ζ ∈ W²^,1((σ_ζ(− ), σ_ζ( ));

R

²) and ¨κ_ζ = −H ˙κ_ζ^⊥. This implies thatκ_ζ is of class C^∞ and the claim follows.

5.

Characterization of convex isoperimetric sets

In this section, we use Theorems 4.1 and 4.2 to characterize convex isoperimetric sets. We also use Theorem A.1 in the appendix on the structure of the characteristic set of bounded convex sets. Here, we recall some known facts about Carnot- Carath´eodory geodesics in the Heisenberg group.

An absolutely continuous path γ : [0, 1] →

H

¹ is said to be horizontal if

˙γ(s) ∈ span_R{X(γ (s)), Y (γ (s))} for a.e. s ∈ [0, 1]. The curve γ = (γ1, γ2, γ3) is horizontal if and only if

γ3(s) = γ3(0) + 2

_s

0

˙γ1γ2− ˙γ2γ1

dσ. (5.1)

The plane curveκ = (γ1, γ2) is the horizontal projection of γ . If κ = (γ1, γ2) is a given plane absolutely continuous curve, then a curveγ = (κ, γ3) with γ3given by (5.1) for someγ3(0) is called a horizontal lift of κ. The standard sub-Riemannian length of the horizontal curveγ is

Length(γ ) =

₁

0

|˙κ(s)| ds, (5.2)

whereκ is the horizontal projection of γ and |˙κ| is the Euclidean length of ˙κ ∈

R

²^. The distance d :

H

¹×

H

¹ → [0, +∞) is defined as the minimum of Length(γ )